Styles of reasoning for mathematics education

Although reasoning is a central concept in mathematics education research, the discipline is still in need of a coherent theoretical framework of mathematical reasoning. With respect to epistemological problems in the dominant discourses on proof, mathematical modelling, and post-truth politics in the discipline, and in accordance with trends in the philosophy of mathematics and in mathematics education research in general, it is argued that it is necessary to give a relativist account of mathematical reasoning. Hacking’s framework of styles of reasoning is introduced as a possible solution. This framework distinguished between at least six different styles of reasoning, many of which are closely connected to mathematics, and argues that these frameworks define what we accept as decidable assertions, as justifications for such assertions, and as possible objects of such assertions. The article ends with a discussion of the implications of the framework for chosen fields of mathematics education research, which may motivate more focussed studies in the future.

Weak Historicism: On Hierarchies of Intellectual Virtues and Goods

This article seeks to reconcile a historicist sensitivity to how intellectually virtuous behavior is shaped by historical contexts with a non-relativist account of historical scholarship. To that end, it distinguishes between hierarchies of intellectual virtues and hierarchies of intellectual goods. The first hierarchy rejects a one-size-fits-all model of historical virtuousness in favor of a model that allows for significant varieties between the relative weight that historians must assign to intellectual virtues in order to acquire justified historical understanding. It grounds such differences, not on the historians’ interests or preferences, but on their historiographical situations, so that hierarchies of virtues are a function of the demands that historiographical situations (defined as interplays of genre, research question, and state of scholarship) make upon historians. Likewise, the second hierarchy allows for the pursuit of various intellectual goods, but banishes the specter of relativism by treating historical understanding as an intellectual good that is constitutive of historical scholarship and therefore deserves priority over alternative goods. The position that emerges from this is classified as a form of weak historicism.